Indirect Dark Matter Detection for Flattened Dwarf Galaxies

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Sep 16, 2016 - Reticulum II signal was consistent with the gamma-ray excess reported from the Galactic Center and claimed. arXiv:1604.05493v4 [astro-ph.

Indirect Dark Matter Detection for Flattened Dwarf Galaxies Jason L. Sanders,1, ∗ N. Wyn Evans,1, † Alex Geringer-Sameth,2, ‡ and Walter Dehnen3, § 1

arXiv:1604.05493v4 [astro-ph.GA] 16 Sep 2016

Institute of Astronomy, Madingley Rd, Cambridge, CB3 0HA 2 McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 3 Department of Physics & Astronomy, University of Leicester, University Road, Leicester, LE1 7RH (Dated: September 19, 2016) Gamma-ray experiments seeking to detect evidence of dark matter annihilation in dwarf spheroidal galaxies require knowledge of the distribution of dark matter within these systems. We analyze the effects of flattening on the annihilation (J) and decay (D) factors of dwarf spheroidal galaxies with both analytic and numerical methods. Flattening has two consequences: first, there is a geometric effect as the squeezing (or stretching) of the dark matter distribution enhances (or diminishes) the J-factor; second, the line of sight velocity dispersion of stars must hold up the flattened baryonic component in the flattened dark matter halo. We provide analytic formulae and a simple numerical approach to estimate the correction to the J- and D-factors required over simple spherical modeling. The formulae are validated with a series of equilibrium models of flattened stellar distributions embedded in flattened dark-matter distributions. We compute corrections to the J- and D-factors for the Milky Way dwarf spheroidal galaxies under the assumption that they are all prolate or all oblate and find that the hierarchy of J-factors for the dwarf spheroidals is slightly altered (typical correction factors for an ellipticity of 0.4 are 0.75 for the oblate case and 1.6 for the prolate case). We demonstrate that spherical estimates of the D-factors are very insensitive to the flattening and introduce uncertainties significantly less than the uncertainties in the D-factors from the other obper cent servables for all the dwarf spheroidals (for example, +10 −3 per cent for a typical ellipticity of 0.4). We conclude by investigating the spread in correction factors produced by triaxial figures and provide uncertainties in the J-factors for the dwarf spheroidals using different physically-motivated assumptions for their intrinsic shape and axis alignments. We find that the uncertainty in the J-factors due to triaxiality increases with the observed ellipticity and, in general, introduces uncertainties of a factor of 2 in the J-factors. We discuss our results in light of the reported gamma-ray signal from the highly-flattened ultrafaint Reticulum II. Tables of the J- and D-factors for the Milky Way dwarf spheroidal galaxies are provided (assuming an oblate or prolate structure) along with a table of the uncertainty on these factors arising from the unknown triaxiality. PACS numbers: 95.35.+d, 95.55.Ka, 12.60.-i, 98.52.Wz

I.

INTRODUCTION

In recent years, gamma-ray observations of Milky Way dwarf spheroidal galaxies (dSphs) have led to great strides in sensitivity to dark matter annihilation. Here the goal is to probe particles which interact with the Standard Model with the well-motivated weak-scale annihilation cross section hσvi ' 3 × 10−26 cm3 s−1 . Particles having this cross section will exist today with an abundance equal to that observed for dark matter ΩDM , making this so-called relic cross section a natural target for experimental searches for annihilation. Combined analyses of dSphs using data from Fermi Large Area Telescope (LAT) first ruled out the relic cross section for dark matter particle masses of a few tens of GeV [1, 2] and followup analyses incorporating more dSphs and increased observation time continue to improve sensitivity [e.g. 3–5]. For higher dark matter masses (M & TeV), the three ma-

∗ Electronic

address: address: ‡ Electronic address: § Electronic address: † Electronic

[email protected] [email protected] [email protected] [email protected]

jor Cherenkov telescope collaborations continue to invest significant time on pointed observations of Milky Way dSphs. The resulting upper limits are two to three orders of magnitude from the relic cross section [6–10], but the situation bodes well for the future CTA project [e.g. 11]. An exciting development in this field is the recent and ongoing discovery of large numbers of new Milky Way satellites made possible by wide-area photometric surveys [e.g. 12–15]. Since 2015 the number of known Milky Way satellites has approximately doubled thanks to Southern hemisphere data from the Dark Energy Survey and Pan-STARRS. These new dSphs have the potential to significantly build on current efforts to uncover evidence of dark matter annihilation [e.g. 16–20]. Intriguingly, the first of these new dwarf spheroidal galaxies discovered, Reticulum II, shows indications of a gamma-ray signal exceeding background in the FermiLAT data [17]. Two methods of modeling the gamma-ray background yield false-alarm probabilities of p = 0.0001 and 0.01 for detecting such a signal. Subsequent analysis [19] confirmed the results of [17] and argued that the Reticulum II signal was consistent with the gamma-ray excess reported from the Galactic Center and claimed

2 as dark matter. With a reprocessing of the raw Fermi data [18], the Fermi-LAT Collaboration found an increased probability for a background fluctuation explaining the Reticulum II signal (p = 0.05) and concluded the signal is insignificant. Making sense of the results of [17] and [18] is complicated by the fact that the two datasets are only partially independent, sharing approximately half the detector events. A separate analysis is needed to compute joint probabilities of background fluctuation in the partially correlated datasets. In this work we follow a different path towards assessing dark matter interpretations of gamma-ray signals. Rather than analyzing the gamma-ray data, we consider the determination of the dark matter content of the Milky Way’s dSphs, a necessary ingredient for performing optimized combined searches using dSphs. A critical test of any alleged dark matter signal from dSphs is that the amplitude of the gamma-ray signal must scale amongst the dSphs according to their J-factors (see, e.g., [21, 22]). The J-factor is the square of the dark matter density integrated along the line of sight and over the solid angle of the observation, Z Z J= ρDM 2 (`, Ω)d`dΩ. (1) While annihilating dark matter models are theoretically better motivated, there are models in which dark matter decays [23]. In these models, the relevant astrophysical factor is the D-factor, which is the dark matter density integrated along the line of sight and over the solid angle of the observation. Robust determinations of the relative J-factors is of prime importance. For instance, the Fermi-LAT Collaboration [18], under the assumption that each of eight considered dSphs was equally likely to produce a signal, further diluted the significance of the Reticulum II gammaray excess to p = 1 − (1 − 0.05)8 = 0.33, concluding that it is insignificant. However, there are reasons to doubt the usefulness of this argument as Reticulum II is closer and very highly flattened, both of which can enhance the amplitude of an annihilation signal compared to other dSphs. Therefore, we require accurate relative estimates of the J- and D-factors, but unfortunately the data on the most tempting dSph candidates are often of limited quality. Motivated by this [24, hereafter Paper I] provided simple formulae for the J- and D-factors for a spherical NFW profile and infinite spherical cusps. The formulae relied on the empirical law that the mass within the half-light radius is well constrained as [25, 26] Mh = M (Rh ) ≈

5 hσ 2 iRh , 2G los

(2)

where Rh is the (projected) half-light radius of the stars 2 and hσlos i is the luminosity weighted squared line-of-sight velocity dispersion. However, an entirely characteristic feature of dSphs is in the name – spheroidal! They are flattened (with a typical ellipticity between 0.3 and 0.5), and some of the

ultrafaints are very highly flattened with ellipticities exceeding 0.5, such as Hercules [27], Ursa Major I [28], Ursa Major II [29], and indeed Reticulum II [30]. Therefore, the underlying physical model of a spherical dark halo containing a round distribution of stars may fail to capture important aspects of the physics. Here we extend the scope of spherical analyses, to account for the effects of flattening in both the stellar and dark matter profiles. Bonnivard et al. [31] provided a systematic investigation of J-factors of flattened figures. Here, two mildly triaxial numerical models of dSphs (created for The Gaia Challenge) were viewed along each of the short, medium and long axes. This investigation revealed that the projection effects can have a significant impact on the velocity dispersion, and concluded that the J-factors constructed by Jeans analyses can vary from the true values by ∼ 2.5. Recently, [32] computed J-factor estimates for the dSphs using axisymmetric Jeans modelling. These authors attributed the differences between their measured J-factors and those from spherical analyses primarily to other modelling assumptions. It is natural to expect that the dissipationless dark matter distribution is rounder – or at least no more flattened – than the dissipative baryonic component. So, large classical dSphs which appear roundish on the sky (such as Leo I and II) may have almost spherical dark matter halos. However, the dark halos of the ultrafaints are expected to be more highly flattened than those of the classical dSphs, as it is known that baryonic feedback effects drive the dark matter distribution towards sphericity [33, 34]. The ultrafaints have such a puny baryonic content that pure dissipationless simulations [35, 36], which find strongly triaxial and nearly prolate dark halos, may be a much better guide to the true shape. For instance, recent simulations have found that the baryonic distribution is just ∼ 10 per cent flatter than the dark-matter distribution for dark-matter halos of 1010 M [37]. Throughout this paper, we work under the assumption that the dark matter distribution is flattened in the same way as the stellar distribution. The effects of flattening can be understood qualitatively for a few simple configurations. The simplest is the face-on case when the dark-matter and stellar distributions are flattened along the line-of-sight. Observationally, the isophotes still appear circular and the measured half-light radius remains the same, but we have increased (decreased) the density of dark matter in the oblate (prolate) case. Naturally, this effect – which we refer to as the geometric factor – gives rise to a larger (smaller) J-factor than a spherical analysis would infer. But, we must also consider the effect of flattening on the line-of-sight velocity dispersion, which we call the kinematic factor. For the oblate case, the stellar distribution is more compressed, so the line-of-sight dispersion is now smaller than the spherically averaged dispersion. Less contained mass is inferred and so the spherical J-factor underestimates the total J-factor. Therefore, for face-on viewing of an oblate figure, both the geometric and kine-

3

II.

MADE-TO-MEASURE FLATTENED EQUILIBRIA

We begin our analysis of the J-factors of flattened dSphs with numerical models constructed by the madeto-measure (M2M) methods [38] as implemented by Dehnen [39]. The models are two-component: dark and stellar. Each component has a target density of the form ρ(m) ∝ p−1 q −1

 m −γ  rs

1+

 m α (γ−β)/α rs

sech

m , (3) rt

x/arcmin

x/arcmin

Oblate

Prolate

10 2 10 1 10 0 10 -1 10 -2 10 -1

10 0

10 1

R/arcmin

10 2 10 5 0

vlos /km s

30 20 10 0 10 20 30 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

y/arcmin

30 20 10 0 10 20 30 10 3

p(vlos )/(km s −1 ) −1

y/arcmin

30 20 10 0 10 20 30 30 20 10 0 10 20 30

M ¯ /arcmin −2

matic effects cause the J-factor inferred from a spherical analysis to be an underestimate of the true value. For the prolate case, the velocity dispersion is larger than the spherically-averaged dispersion and so more mass is inferred and the spherical J-factor is an overestimate. When the dSph is viewed edge-on such that it appears flattened in the sky, the combined result of the kinematic and geometric effects is less clear. For oblate figures, the density is increased over the spherical case, whilst the half-light radius remains the same. These geometric effects cause the J-factor assuming sphericity to be an underestimate. However, the kinematic factor works the other way, as the measured velocity dispersion is greater than the spherical average. We will see that the combination of these two competing effects leads to a small decrease in the true J-factor over that inferred from a spherical analysis. For the prolate case, we have the converse situation with the geometric factor leading to an overestimate whilst the kinematic factor leads to an underestimate. However, now the stretching of the stellar profile in the sky causes the half-light radius to increase. We will see that the net result is an increase in the true J-factor over the spherical J-factor. This qualitative explanation is tested in Section II where we construct equilibrium models of the Reticulum II galaxy via the made-to-measure method. We explore a range of different flattenings and provide simple fits for the correction factors. In Section III, we use these fits to derive J-factors for the known dwarf spheroidals under the assumption that they are either prolate or oblate. In Section IV, we build intuition for our numerical results by considering two families of axisymmetric equilibria for which analytic progress is possible and present a more rapid general approach for estimating the correction factors using the virial theorem. Section V extends these findings to the triaxial case and demonstrates how the correction factors vary for a triaxial figure as a function of the viewing angle. In Section VI, we discuss the constraints and evidence on the intrinsic shapes and alignments of the Milky Way dSphs and give estimates of the uncertainties in the J-factors of the dSphs due to unknown triaxiality. In Section VII we summarize our findings and discuss possible implications for the claimed signal from Reticulum II in light of our work.

5 10 15 −1

FIG. 1: Reticulum II M2M equilibria of a flattened Plummer distribution of stars in a flattened NFW dark halo. The top left panel shows the logarithm of the projected mass distribution of an oblate model viewed edge-on with axis ratio 0.4. The contours are logarithmically spaced. The top right panel shows the logarithm of the projected mass distribution of a prolate model viewed edge-on with axis ratio 0.4. Note the ‘X’-shape in the prolate case. The bottom left panel shows the surface density profiles in elliptical bins with Plummer profile fits (oblate in blue, prolate in dashed green). The bottom right panel shows the line-of-sight velocity distributions (oblate in blue, prolate in dashed green).

where m2 = x2 + (y/p)2 + (z/q)2 . This is the familiar double power-law with scale radius rs , with an exponential taper at the tidal radius rt . For r  rs , the density falls like r−γ , whilst for r  rs , it falls like r−β . The case α = 1, β = 3, γ = 1 is the NFW dark halo. Plummer models are often used to describe the light profiles of dSphs [see e.g., 40, 41]. They correspond to the parameters α = 2, β = 5, γ = 1 and rt = ∞. We begin by constructing two flattened spheroidal (p = 1) models of the Reticulum II dSph. For both models, the dark halo is a NFW model (α = 1, β = 3, γ = 1, rs = 1, rt = 10). The stars follow a Plummer profile (α = 2, β = 5, γ = 0, rs = 0.5, rt = 9). The chosen ratio of the dark matter scale radius to the stellar scale radius lies within the measured range for the Local Group dSphs (∼ 1.25 to ∼ 30) [42]. The two models differ in their shape. The first model is oblate in both the stars and the dark matter with an axis ratio of q = 0.4 (chosen to match the observed axis ratio of Reticulum II of 0.39 [30]). The second model is prolate with an axis ratio of q = 2.5. When viewed along the x-axis both models appear flattened with axis ratio 0.4. In addition, we construct a third spherical model as a reference. This has the same parameters, but without the flattening in either

4 the dark matter or the stars. The dark NFW halos source the potential (computed using a bi-orthonormal basis expansion [39]) in which the weights of the Plummer models are adjusted until the target densities are reached. No other constraints on the distribution functions are used. We use a 107 particle realization of the flattened NFW distribution to compute the potential. The constraints on the Plummer model are generated with 100 realizations of 106 particles and 106 particles are used in the M2M simulation. To check convergence, the models were run turning off the weight adjustment in the M2M code. Both flattened models exhibit a slow drift in the density constraint suggesting they are not perfect equilibrium models. However, this is almost certainly true for the actual dSphs which reside in the tidal field of the Milky Way. Reticulum II has a half-light major axis length of 5.63 arcmin, is at a distance of ∼ 30 kpc [30] and has a line-of-sight velocity dispersion of 3.22 km s−1 [43]. To match the final models to the observed constraints on Reticulum II, we compute the projected half-light major axis length (fitted with a Plummer model) and the projected line-of-sight velocity dispersion. We then compute the scale factors R and V that scale the radial distributions and the velocity distributions to the observations. The corresponding total mass of the dark matter profile (set to unity in the simulation) is then scaled by a factor M = RV 2 . For the spherical model we match the halflight major axis length to an ‘ellipticity-corrected’ radius given by the geometric mean of the half-light major and minor axis lengths. This is related√to the observed halflight major axis length Rh as Rh 1 −  where  is the ellipticity. In Figs. 1, we show the final projected distributions of the two flattened models. Note that for the prolate case, the models do not completely reproduce the target density profile as there is a clear ‘X’ shape in the (x, y) plane. Additionally, we show the surface density of the two models (using a mass-to-light ratio of 500, [43]) and the line-of-sight velocity distributions. The prolate velocity distribution is slightly peakier than the oblate case but such a small difference would not be detectable observationally. To explore the effects of adjusting the stellar and darkmatter profiles, we also build two further models, one with a central cusp in the stellar profile (γ = 1) and one with a cored dark-matter profile with parameters α = 1, β = 4 and γ = 0.

A.

source is given by1 Z +∞ Z Dθ Z 2π 1 dz dR R dφ ρ2DM , J(θ) = 2 D −∞ 0 0

(4)

where D is the distance to the source (30 kpc for Reticulum II) and θ is the beam angle. Similarly, the D-factor is given by Z +∞ Z Dθ Z 2π 1 D(θ) = 2 dz dR R dφ ρDM . (5) D −∞ 0 0 In Table I we report the J- and D-factors at θ = 0.5◦ (the typical observational resolution). We also show the J- and D-factors for the spherical model computed from the formulae of Paper I. We see that these formula underestimate the J-factor by a factor 1.2 and the D-factor by a factor 1.05. We also record the correction factor between the prolate / oblate models and the spherical models using the notation FJ = log10 (J/Jsph ), FD = log10 (D/Dsph ).

(6)

The oblate model with NFW dark matter and Plummer light has a J-factor that is diminished by a factor 1.4 over the spherical model and a D-factor that is diminished by a factor 1.3. On the other hand, the prolate model has an enhancement in the J-factor by a factor 3.4 and a small decrease in the D-factor of 10 per cent. The near-prolate model with a cuspy stellar profile produces a very similar J-factor to the Plummer prolate model, but here the D-factor is enhanced over the spherical model by 20 per cent. Finally, in a similar fashion to the prolate NFW profile, the prolate cored dark matter profile also produces an enhancement in the J-factor of a factor 3 and a small diminution in the D-factor of order 10 per cent. B.

A Range of Flattenings

We have established that a prolate model of Reticulum II viewed edge-on produces a significant enhancement in the J-factor over its spherical counterpart, whilst an oblate model has a slight diminution. However, the observed dSphs span a whole range of ellipticities, so we now go on to explore models with a variety of flattenings. We construct 3 oblate M2M models with the same parameters as the spherical reference model in Table I but with flattenings q = 0.5, 0.6, 0.7, and similarly 3 prolate M2M models with flattenings q = 1.423, 1.667, 2. Again the M2M models are normalized to match the line-ofsight velocity dispersion and half-light major-axis length of Reticulum II.

J- and D-factors 1

For our five models of Reticulum II, we proceed to calculate the J- and D-factors. The J-factor for a distant

When computing these integrals numerically, we have found it useful to perform the coordinate transformation tan χ = z/rs where rs is the scale radius of the density profile.

5 −5

TABLE I: J- and D-factors for a beam angle of 0.5◦ for a series of Reticulum II models. The J-factors are in units of GeV2 cm and the D-factors are in units of GeV cm−2 . Each model was normalized such that the line-of-sight velocity dispersion and half-light major axis length matched that of Reticulum II. For the spherical model, an ‘ellipticity-corrected’ half-light radius of √ Rh 1 −  (where  is the ellipticity) was used to scale the models. Note the correction factors are with respect to the spherical NFW, spherical Plummer model in the first row not with respect to the corresponding spherical model. Model Spherical NFW Spherical Plummer

Paper I J Paper I D log10 (J(0.5◦ )) log10 (D(0.5◦ ))

FJ

FD

18.56

17.56

18.64

17.58

0.00 0.00

Oblate NFW, p = 1, q = 0.4 Oblate Plummer, p = 1, q = 0.4

-

-

18.45

17.62

−0.19 0.05

Prolate NFW, p = 1, q = 2.5 Prolate Plummer, p = 1, q = 2.5

-

-

19.05

17.67

0.40 0.09

Near-prolate NFW, p = 0.5, q = 0.4 Near-prolate cuspy Plummer α? = 2, β? = 5, γ? = 1, p = 0.4, q = 0.38

-

-

19.01

17.78

0.37 0.20

Prolate cored DM αDM = 1, βDM = 4, γDM = 0, p = 1, q = 2.5 Prolate Plummer, p = 1, q = 2.5

-

-

18.90

17.66

0.26 0.08

The J- and D-factors for our series of models are plotted in Fig. 2. All models are viewed such that they appear maximally flattened (along the short-axis for the prolate cases and along the long-axis for the oblate cases). We also show the J- and D-factors computed using the simple formulae (equations 15 and 19) from Paper I. We see that this formula disagrees with the spherical case by ∼ 0.2 due to the use of the empirical relation for the half-light mass. As shown in Paper I, for most dSphs this is less than the uncertainty in the J-factor due to uncertainties in the line-of-sight velocity dispersion and half-light radius. The prolate models produce a sequence of more enhanced J-factor at all angles as we increase the flattening q. The oblate models produce a similar sequence of decreasing J as we decrease the flattening q. These trends are reproduced in the D-factor. Note the asymmetry with q in both J and D: the equivalent flattening for an prolate model produces a larger difference from the spherical model than the corresponding oblate model. With this sequence of models, we also investigate how the J-factor for an apparently round dSph changes as the dSph is flattened along the line of sight. In Fig. 3, we show the range of J- and D-factors for the set of flattened models viewed face-on such that the isophotes appear round and all models have the same half-light radius. We see that the range of possible J-factors with flattening along the line of sight varies by a factor of 10. The oblate models all have a similar decrease in the J-factor. The D-factor is unaffected by flattening along the line of sight. For the series of flattened M2M models, we compute the correction factors FJ and FD by comparing each model with the spherical model with the same line-ofsight velocity dispersion and the ‘ellipticity-corrected’

TABLE II: Slopes η of the base-10 logarithms of the correction factors with respect to log10 q fitted to the made-to-measure models of Section II. The prolate and oblate cases are treated separately. The multiplicative factor by which a J- or Dfactor from a spherical analysis must be corrected is given by q η . Note the spherical models to which we√compare use an ‘ellipticity-corrected’ half-light radius of Rh 1 −  where  = 1 − q is the ellipticity in the oblate case and  = 1 − 1/q in the prolate case. View η Oblate (q < 1) η Prolate (q > 1) FJ Edge-on 0.534 0.899 Face-on -1.647 -1.181 FD Edge-on 0.056 0.177 Face-on -0.335 -0.089

half-light radius. The trends of FJ and FD with respect to q are very smooth so we opt to fit the corrections from the models with a simple functional form

Ffit = η log10 (q)

(7)

where we fit q < 1 and q > 1 separately. The values of η chosen are given in Table II. Although our fit is an extrapolation for q < 0.4 and q > 2.5, we will see that it agrees well with the more involved models of Section IV. In reality, the correction factors are a function of the beam angle. We have found that the correction factors are very insensitive to the beam angle so this formula is appropriate for all dSphs irrespective of their size compared to the resolution of the instrument.

6

q

q 19.5

log 10 (J(θ)/ GeV 2 cm −5 )

19.2 19.0 18.8 18.6 18.4 18.2 18.0 17.8 18.0

Edge-on

19.0 18.5 18.0 17.5

Face-on

18.0

17.5 17.0 16.5 16.0 15.5 15.0

17.5 17.0 16.5 16.0 15.5 15.0

0.0

0.1

0.2

0.3

0.4

0.5

θ/ deg FIG. 2: J- and D-factors as a function of beam angle for a range of flattened models viewed edge-on with identical lineof-sight velocity dispersions and half-light major-axis lengths. The models are coloured by the flattening in the density of both the stars and dark matter, q. The spherical model is shown with the short-dashed line, whilst the analytic formula for the NFW model (equations (15) and (19) from Paper I) is shown with the long-dashed line.

III.

0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

log 10 (D(θ)/ GeV cm −2 )

log 10 (D(θ)/ GeV cm −2 )

log 10 (J(θ)/ GeV 2 cm −5 )

0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

J- AND D-FACTORS FOR THE MILKY WAY DSPHS

We now apply the corrections to the J- and D-factors of the observed dSphs. They are listed in Table III along with their measured ellipticities  = 1 − b/a where b/a is the observed axis ratio. We take the majority of the ellipticities and ±1σ error-bars from the review of [44]. The ellipticities of the new dSphs discovered in the Dark Energy Survey are taken from [30], the ellipticity of Pisces II is taken from [45] and that of Hydra II from [46]. For both Leo T and Horologium I, only upper-bounds on the ellipticity are available. For each dSph, we compute the correction factor assuming the dSph is either oblate or prolate and observed edge-on. We draw samples from the error distributions of the ellipticities and compute the median and ±1σ values of the correction factors for both the J- and D-factors using equation (7). The baseline spherical model to which we √are comparing uses an effective half-light radius of Rh 1 − . We combine these estimates with the spheri-

0.0

0.1

0.2

0.3

0.4

0.5

θ/ deg FIG. 3: J- and D-factors as a function of beam angle for a range of flattened models viewed face-on with identical lineof-sight velocity dispersions and half-light major-axis lengths. The models are coloured by the flattening in the density of both the stars and dark matter, q. The spherical model is shown with the short-dashed line and the model using the formulae from Paper I is shown with the long-dashed line.

cal estimates computed in Paper I (adding the errors in quadrature). The results of this procedure are reported in Tables III and IV. We show this data in Fig. 4. We plot the distribution of J- and D-factors for the dSphs assuming they are spherical, oblate or prolate. Ursa Major I has the largest ellipticity and hence the largest prolate correction factor (a factor of ∼ 4). Reticulum II, Ursa Major II and Hercules all have ellipticities ∼ 0.6 and so the prolate correction factors are approximately ∼ 2.2 − 2.7. For ellipticities less than ∼ 0.4, the correction factors are less than the errors on the spherical J-factors. For every dSph the correction to the D-factors is smaller than the errors in the spherical D-factor. Hence, we conclude that flattening has a negligible effect on the D-factor estimates. If the entire population of dSphs is prolate then only Tucana II and Willman 1 have potentially higher Jfactors than Reticulum II, with both Ursa Major II and Segue 1 having a very similar J-factor to Reticulum II. We remark that Tucana II is consistent with having circular isophotes [30], whilst the assumption of dynamical equilibrium for Willman 1 is dubious [47]. Similarly, Ursa

7 Major II appears to be in process of severe tidal disruption [29]. Finally, the J-factor of Segue 1 has been shown to be extremely sensitive to the presence of foreground contaminants [e.g., 48, 49]. These final three dSphs have been marked in red in Figure 4 to indicate their dubious J-factors. Therefore, it is possible that the Reticulum II gamma-ray signal may be due to annihilation if the dwarf has a prolate shape. We can robustly conclude from Figure 4 that if Reticulum II has a prolate shape then an observed annihilation signal from only Reticulum II is not in tension with the lack of signals from all the other dSphs irrespective of their shapes. If, however, Reticulum II is oblate and has an observed annihilation signal we begin to have some tension if there is a lack of signal from the other dSphs. The majority of this tension arises from those problematic dSphs already mentioned. However, if both Ursa Minor and Tucana II have prolate shapes it becomes unlikely that they both have smaller J factors than an oblate Reticulum II.

1. For a given viewing angle (ϑ, ϕ), we find the measured ellipticity and orientation of the observed minor axis (using, for instance, equations (A1,A2,A6) of [50]) and compute the elliptical half-light radius Rh0 . This gives us a length scaling R = Rh /Rh0 , and encodes the geometric factor described in the introduction. 2. In principle, to solve for the kinematics of the stars in the dark matter potential, we could use the axisymmetric Jeans equations. However, there are degeneracies in the solution space and only a few algorithms exist for solution [51, 52]. As we need only match an integrated quantity, we use the virial 2 theorem to compute hσlos i as 02 hσlos i=

SEMI-ANALYTIC MODELS

Numerical M2M models provide a robust method for determining the corrections required when modeling flattened systems as spherical. However, they are computationally expensive to construct so cannot be employed in a Markov Chain Monte Carlo analysis that requires many models. We have provided a simple fitting formula for our model setup, but there will be some variation in the correction factors depending on, for instance, the light profile, the density profile of the dark matter, and the ratio of the scale lengths of the light to the dark matter. We now proceed to understand and reproduce the results of the M2M models using simpler methods. In subsection A, we describe a general virial method to compute J-factors for flattened halo models. This numerical algorithm can be applied to any dark matter density, but in the two following subsections, we provide analytic shortcuts to evaluate the J-factors for two specific families – the infinite flattened cusps and the flat rotation curve halos. Readers primarily interested in the results, rather than the details of the methods, should skip to sub-section D, where we compare our models to the M2M results. Figures 6 and 7 provide summary plots, which show the range of correction factors as a function of the flattening of the stellar density.

A.

The Virial Method

We can construct approximate equilibrium models much more cheaply than with the full M2M apparatus by using the virial theorem. The two constraints provided by the data are the integrated line-of-sight velocity 2 dispersion hσlos i and the half-light major-axis length Rh . We describe a method to match these observations given density models for the light (ρ? ) and dark matter (ρDM ).

(8)

where Z

IV.

Wlos 2 Wlos hσtot i = , W M

Wlos =

d3 x ρ? Rij xj

∂ΦDM Rki . ∂xk

(9)

ΦDM is the dark matter potential (generically computed using a multipole expansion [53]), M is the total dark matter mass and Rij is the projection matrix along the line-of-sight from coordinates aligned with the principal axes of the dSph. We have used the Einstein summation convention. For triaxial symmetry, the cross-terms in the integral vanish so we need only project the velocity dispersions along the principal axes. This gives us a ve2 02 locity scaling V = hσlos i/hσlos i, and encodes the kinematic factor described in the introduction. 3. We compute a mass scaling M = V 2 R. The initial model is scaled by M and R and the J- and D-factors are computed. These can be compared to the spherical model with the same line-of-sight velocity dispersion and half-light radius. This algorithm is completely general. For some special choices of stellar and dark matter density, the integration in the virial theorem can be performed analytically. We now give two examples – infinite flattened cusps and flat rotation curve halos – for which the virial integrals can be done. This means that the behavior of the J-factor at fixed observables (line of sight velocity dispersion and half-light radii) can be mapped out analytically as a function of flattening or concentration.

B.

Flattened Cusps

Let us take the dark matter halo as an axisymmetric cusp stratified on similar concentric spheroids with an axis ratio q. If the cusps have the same mass Mh within the spheroidal half-light radius mh , then the mass

Name Hydra II Leo T Leo II Segue 2 Leo I Horologium I Fornax Draco Sculptor Carina Sextans Coma Berenices Bo¨ otes I Canes Venatici I Tucana II Pisces II Grus I Willman 1 Segue 1 Leo IV Leo V Canes Venatici II Ursa Minor Reticulum II Ursa Major II Hercules Ursa Major I

 log10 (Jsph (0.5◦ )) +0.20 0.01−0.01 16.56+0.87 −1.85 < 0.10 17.32+0.38 −0.37 0.13+0.05 17.44+0.25 −0.05 −0.25 0.15+0.10 17.11+0.85 −0.10 −1.76 0.21+0.03 17.80+0.28 −0.03 −0.28 < 0.28 18.64+0.95 −0.39 0.30+0.01 18.15+0.16 −0.01 −0.16 0.31+0.02 18.86+0.24 −0.02 −0.24 +0.03 0.32−0.03 18.65+0.29 −0.29 0.33+0.05 17.99+0.34 −0.05 −0.34 0.35+0.05 17.87+0.29 −0.05 −0.29 0.38+0.14 18.67+0.33 −0.14 −0.32 0.39+0.06 16.65+0.64 −0.06 −0.38 0.39+0.03 17.27+0.11 −0.03 −0.11 0.39+0.10 19.05+0.87 −0.20 −0.58 0.40+0.10 17.90+1.14 −0.10 −0.80 0.41+0.20 17.96+0.90 −0.28 −1.93 0.47+0.08 19.29+0.91 −0.08 −0.62 0.48+0.13 19.41+0.39 −0.13 −0.40 +0.11 0.49−0.11 16.64+0.90 −0.90 16.94+1.05 0.50+0.15 −0.72 −0.15 0.52+0.11 17.65+0.40 −0.11 −0.40 0.56+0.05 19.15+0.25 −0.05 −0.24 0.59+0.02 18.71+0.84 −0.03 −0.32 0.63+0.05 19.38+0.39 −0.05 −0.39 0.68+0.08 16.83+0.45 −0.08 −0.45 0.80+0.04 18.48+0.25 −0.04 −0.25 Oblate FJ −0.002+0.002 −0.052 −0.012+0.008 −0.009 −0.032+0.011 −0.017 −0.037+0.019 −0.042 −0.055+0.008 −0.010 −0.034+0.024 −0.028 −0.083+0.003 −0.003 −0.086+0.006 −0.007 −0.090+0.010 −0.011 −0.093+0.016 −0.020 −0.100+0.016 −0.020 −0.108+0.039 −0.071 −0.114+0.020 −0.026 −0.115+0.011 −0.012 −0.114+0.053 −0.049 −0.119+0.032 −0.048 −0.117+0.064 −0.117 −0.147+0.030 −0.040 −0.151+0.045 −0.076 −0.156+0.041 −0.064 −0.159+0.053 −0.096 −0.171+0.042 −0.065 −0.191+0.024 −0.029 −0.207+0.016 −0.012 −0.231+0.028 −0.035 −0.266+0.050 −0.069 −0.373+0.041 −0.053

Prolate FJ log10 (Jobl (0.5◦ )) log10 (Jpro (0.5◦ )) 0.004+0.087 16.56+0.87 16.56+0.87 −0.003 −1.85 −1.85 +0.38 0.020+0.015 17.31 17.34+0.38 −0.014 −0.37 −0.37 0.054+0.029 17.41+0.25 17.49+0.25 −0.018 −0.25 −0.25 0.063+0.070 17.07+0.85 17.17+0.85 −0.032 −1.76 −1.76 0.092+0.017 17.75+0.28 17.89+0.28 −0.014 −0.28 −0.28 0.058+0.047 18.61+0.95 18.70+0.95 −0.040 −0.39 −0.39 0.139+0.006 18.07+0.16 18.29+0.16 −0.005 −0.16 −0.16 0.145+0.012 18.77+0.24 19.00+0.24 −0.011 −0.24 −0.24 +0.019 +0.29 0.151−0.016 18.56−0.29 18.80+0.29 −0.29 0.156+0.034 17.90+0.34 18.15+0.34 −0.026 −0.34 −0.34 0.169+0.034 17.77+0.29 18.04+0.29 −0.028 −0.29 −0.29 0.182+0.120 18.56+0.33 18.85+0.35 −0.066 −0.33 −0.33 0.192+0.043 16.54+0.64 16.84+0.64 −0.033 −0.38 −0.38 0.194+0.020 17.15+0.11 17.46+0.11 −0.018 −0.11 −0.11 0.193+0.082 18.94+0.87 19.24+0.87 −0.090 −0.58 −0.59 0.200+0.081 17.78+1.14 18.10+1.14 −0.053 −0.80 −0.80 0.198+0.196 17.84+0.90 18.16+0.92 −0.107 −1.93 −1.93 0.247+0.067 19.14+0.91 19.54+0.91 −0.051 −0.62 −0.62 0.255+0.128 19.26+0.39 19.66+0.41 −0.076 −0.41 −0.41 +0.108 +0.90 0.262−0.068 16.48−0.90 16.90+0.91 −0.90 0.268+0.161 16.78+1.05 17.21+1.06 −0.089 −0.73 −0.73 0.288+0.109 17.48+0.40 17.94+0.41 −0.071 −0.41 −0.41 0.321+0.049 18.96+0.25 19.47+0.25 −0.040 −0.24 −0.24 0.348+0.020 18.50+0.84 19.06+0.84 −0.027 −0.32 −0.32 0.389+0.060 19.15+0.39 19.77+0.39 −0.047 −0.39 −0.39 0.447+0.116 16.56+0.45 17.28+0.46 −0.084 −0.46 −0.46 0.629+0.089 18.11+0.25 19.11+0.27 −0.070 −0.26 −0.26

TABLE III: Annihilation correction factors for dwarf spheroidals due to their observed ellipticity  (note Leo T and Horologium I only have upper-bounds on the −5 ellipticity). We report the spherical J-factor for a beam angle of 0.5◦ in units of GeV2 cm along with the corrections FJ assuming the galaxy is observed exactly edge-on and is either oblate or prolate. We report the resultant J-factors for these cases as Jobl and Jpro in units of GeV2 cm−5 . The dSphs are ordered by their ellipticity.

8

Name Hydra II Leo T Leo II Segue 2 Leo I Horologium I Fornax Draco Sculptor Carina Sextans Coma Berenices Bo¨ otes I Canes Venatici I Tucana II Pisces II Grus I Willman 1 Segue 1 Leo IV Leo V Canes Venatici II Ursa Minor Reticulum II Ursa Major II Hercules Ursa Major I

 log10 (Dsph (0.5◦ )) +0.20 0.01−0.01 16.89+0.44 −0.92 < 0.10 17.35+0.37 −0.37 0.13+0.05 17.62+0.25 −0.05 −0.25 0.15+0.10 17.08+0.86 −0.10 −1.75 0.21+0.03 17.89+0.28 −0.03 −0.28 < 0.28 17.78+0.47 −0.20 0.30+0.01 18.26+0.17 −0.01 −0.17 0.31+0.02 18.39+0.25 −0.02 −0.25 +0.03 +0.29 0.32−0.03 18.33−0.29 0.33+0.05 17.98+0.34 −0.05 −0.34 +0.05 0.35−0.05 18.07+0.29 −0.29 18.06+0.32 0.38+0.14 −0.32 −0.14 0.39+0.06 17.28+0.64 −0.06 −0.38 17.78+0.11 0.39+0.03 −0.11 −0.03 0.39+0.10 18.45+0.88 −0.20 −0.58 0.40+0.10 17.41+0.57 −0.10 −0.40 0.41+0.20 17.59+0.46 −0.28 −0.96 0.47+0.08 18.03+0.91 −0.08 −0.62 0.48+0.13 18.17+0.39 −0.39 −0.13 +0.11 +0.90 0.49−0.11 17.22−0.90 0.50+0.15 17.23+1.05 −0.15 −0.70 +0.11 0.52−0.11 17.37+0.40 −0.40 0.56+0.05 18.45+0.24 −0.05 −0.24 0.59+0.02 17.93+0.85 −0.03 −0.32 18.48+0.39 0.63+0.05 −0.05 −0.39 0.68+0.08 17.38+0.45 −0.08 −0.45 0.80+0.04 18.15+0.25 −0.04 −0.25 Oblate FD −0.000+0.000 −0.005 −0.001+0.001 −0.001 −0.003+0.001 −0.002 −0.004+0.002 −0.004 −0.006+0.001 −0.001 −0.004+0.002 −0.003 −0.009+0.000 −0.000 −0.009+0.001 −0.001 −0.009+0.001 −0.001 −0.010+0.002 −0.002 −0.011+0.002 −0.002 −0.011+0.004 −0.007 −0.012+0.002 −0.003 −0.012+0.001 −0.001 −0.012+0.006 −0.005 −0.012+0.003 −0.005 −0.012+0.007 −0.012 −0.015+0.003 −0.004 −0.016+0.005 −0.008 −0.016+0.004 −0.007 −0.017+0.006 −0.010 −0.018+0.004 −0.007 −0.020+0.002 −0.003 −0.022+0.002 −0.001 −0.024+0.003 −0.004 −0.028+0.005 −0.007 −0.039+0.004 −0.006

Prolate FD log10 (Dobl (0.5◦ )) log10 (Dpro (0.5◦ )) 0.001+0.017 16.89+0.44 16.89+0.44 −0.001 −0.92 −0.92 +0.37 0.004+0.003 17.35 17.35+0.37 −0.003 −0.37 −0.37 0.011+0.006 17.62+0.25 17.63+0.25 −0.004 −0.25 −0.25 0.012+0.014 17.08+0.86 17.09+0.86 −0.006 −1.75 −1.75 0.018+0.003 17.88+0.28 17.91+0.28 −0.003 −0.28 −0.28 0.011+0.009 17.78+0.47 17.79+0.47 −0.008 −0.20 −0.20 0.027+0.001 18.25+0.17 18.29+0.17 −0.001 −0.17 −0.17 0.029+0.002 18.38+0.25 18.42+0.25 −0.002 −0.25 −0.25 +0.004 +0.29 0.030−0.003 18.32−0.29 18.36+0.29 −0.29 0.031+0.007 17.97+0.34 18.01+0.34 −0.005 −0.34 −0.34 0.033+0.007 18.06+0.29 18.10+0.29 −0.005 −0.29 −0.29 0.036+0.024 18.05+0.32 18.10+0.32 −0.013 −0.32 −0.32 0.038+0.008 17.27+0.64 17.32+0.64 −0.007 −0.38 −0.38 0.038+0.004 17.77+0.11 17.82+0.11 −0.004 −0.11 −0.11 0.038+0.016 18.44+0.88 18.49+0.88 −0.018 −0.58 −0.58 0.039+0.016 17.40+0.57 17.45+0.57 −0.010 −0.40 −0.40 0.039+0.039 17.58+0.46 17.63+0.46 −0.021 −0.96 −0.96 0.049+0.013 18.01+0.91 18.08+0.91 −0.010 −0.62 −0.62 0.050+0.025 18.15+0.39 18.22+0.39 −0.015 −0.39 −0.39 +0.021 +0.90 0.052−0.013 17.20−0.90 17.27+0.90 −0.90 0.053+0.032 17.21+1.05 17.28+1.05 −0.018 −0.70 −0.70 0.057+0.022 17.35+0.40 17.43+0.40 −0.014 −0.40 −0.40 0.063+0.010 18.43+0.24 18.51+0.24 −0.008 −0.24 −0.24 0.069+0.004 17.91+0.85 18.00+0.85 −0.005 −0.32 −0.32 0.077+0.012 18.46+0.39 18.56+0.39 −0.009 −0.39 −0.39 0.088+0.023 17.35+0.45 17.47+0.45 −0.017 −0.45 −0.45 0.124+0.018 18.11+0.25 18.27+0.25 −0.014 −0.25 −0.25

TABLE IV: As Table III, but for the decay correction factors. The D-factors are quoted in units of GeV cm−2 .

9

10

log 10 (J(0. 5 ◦ )/ GeV 2 cm −5 )

22

Spherical

21

Oblate

Prolate

20

Reticulum II +1σ prolate

19

Reticulum II +1σ oblate

18 17 16 15 14

4 2

1 1

7 6

2 3

8 9

6 10 11 5 13 15 14 12 9 16 17 18 22 21 25 20 19 27 26 23 24 5 11 12 7 8 15 14 13 10 17 19 16 22 24 25 20 18 23 26 21 27

Segue 1 Ursa Major II Willman 1 Ursa Minor Tucana II Draco Reticulum II Coma Berenices Sculptor Horologium I Ursa Major I Fornax Carina Grus I Pisces II Sextans Leo I Canes Venatici II Leo II Leo T Canes Venatici I Segue 2 Leo V Hercules Boötes I Leo IV Hydra II

13

3 4

20

log 10 (D(0. 5 ◦ )/ GeV cm −2 )

Spherical

Oblate

Prolate

19 18 17 16

Segue 1 Ursa Major II Willman 1 Ursa Minor Tucana II Draco Reticulum II Coma Berenices Sculptor Horologium I Ursa Major I Fornax Carina Grus I Pisces II Sextans Leo I Canes Venatici II Leo II Leo T Canes Venatici I Segue 2 Leo V Hercules Boötes I Leo IV Hydra II

15

FIG. 4: J- (top) and D-factors (bottom) integrated over a beam angle of 0.5◦ for 27 dSphs. The diamonds with red error-bars are computed assuming a spherical model and are taken from Paper I. The circles with blue error-bars show the spherical J-factors adjusted by the oblate correction factors marginalized over the uncertainty in the ellipticity (assuming the galaxy is observed edge-on) and the squares with black error-bars show the spherical J-factors adjusted by the prolate correction factors marginalized over the uncertainty in the ellipticity (assuming the galaxy is observed edge-on). The dSphs are ordered by their median spherical J-factors. The top set of red numbers gives the ordering of the upper-limits on the spherical J-factors, and the bottom set of black numbers gives the ordering of the upper-limits on the prolate J-factors. The gray dashed lines show the 1σ upper-limit for the Reticulum II assuming it is prolate or oblate. The three dSphs with red names have unknown additional systematic uncertainties due to the presence of contaminants or the questionable assumption of dynamical equilibrium.

11 enclosed is  M (m) = Mh

m mh

3−γDM for m ≤ rt

(10)

and M = Mh (rt /mh )3−γDM otherwise. m2 = x2 + y 2 + z 2 q −2 = R2 +z 2 q −2 and rt is a hard truncation ellipsoidal radius. The dark matter density is ρDM (m) =

Mh 3 − γDM DM mγDM 4πqm3−γ h

for m ≤ rt ,

(11)

and zero otherwise. Note the factor of q in the denominator which comes from the Jacobian. It means that the oblate models (q < 1) in the sequence have an increased density as compared to their spherical progenitor, whilst the prolate models (q > 1) have a decreased density. The spherical member of the family obeys the empirical law (2). As the mass Mh is preserved along the sequence, we can still use Eq. (2) for the flattened cusps provided we correct the observables – the line of sight velocity dispersion and the projected half-light radius – to the spherical parent. For comparison purposes, it is useful to define the Jfactor and D-factor of the infinite spherical cusp (rt → ∞, equations (8) and (11) in Paper I) as Jsph =

2 1  hσlos iRh 2  Dθ 3−2γDM P (γDM ), (12) D2 Rh3 G Rh

Dsph

1 hσ 2 iRh  Dθ 3−γDM = 2 los Q(γDM ), D G Rh

(13)

(14)

If an oblate model is viewed along the short axis, or a prolate model is viewed along the long axis, then it appears round. The line of sight coincides with the symmetry or z axis. The geometric corrections are then straightforward to evaluate as Jgeo,face =

1 , q

Dgeo,face = 1.

Jgeo,edge =

q 2−γDM 2πq 2



Z

dθ(cos2 θ + q −2 sin2 θ)1/2−γDM ,

0

(16) and Dgeo,edge =

q 1−γDM /2 2πq

Z

(15)

This case is very simple because both the field of view and the surface density contours are circular. Note that both factors are independent of the slope of the density profile γDM . If an infinite (rt → ∞) oblate or prolate model is viewed edge-on, it appears flattened with axis ratio q. The line of sight then coincides with, say, the y direction. Observationally, the effective radius of a flattened



dθ(cos2 θ+q −2 sin2 θ)1/2−γDM /2 .

0

(17) The factors can only be reduced to a single quadrature due to the mismatch between the circular beam aperture and the elliptical isophotes. For oblate (prolate) models, the geometric correction leads to an increase (decrease) in the J-factor as compared to a spherical model with the same Mh if γDM ≤ 2. If the dark matter halo is truncated at a finite ellipsoidal radius rt < Dθ, the beam encloses all the dark matter and the edge-on geometric factors reduce to Jgeo,edge = q 1−γDM ,

where both P (γDM ) and Q(γDM ) are constants given in Paper I. In these expressions, the half-light radius Rh is the ‘ellipticity-corrected’ half-light radius that includes a √ factor of 1 − . The J- and D- factors for our axisymmetric models can now be written in the form J = Jsph Jgeo Jkin , D = Dsph Dgeo Dkin .

model is always measured along the projected major axis. For an oblate model, the measured effective radius is Rh whereas for the prolate model, the effective radius of its spherical progenitor is actually Rh /q. Additionally, comparison with the ‘ellipticity-corrected’ √ spherical model gives rise to an additional factor of 1 −  in the √ effective radius which equals q for the oblate case and p 1/q for the prolate case. The geometric corrections (i.e. the ratio of the J- and D-factors to those for a spherical model with the same Mh ) are now

Dgeo,edge = q 1−γDM /2 .

(18)

These equations are preferable as for γDM < 3 they correspond to finite mass models and for γDM < 3/2 they produce finite J-factors. We have found that they give much better representations of the correction factors for more general models. We have computed the ratio of the J- and D-factors to those of the spherical model with the same Mh . As Mh is estimated from the line-of-sight velocity dispersion, we must now compute the ratio of the true Mh to that computed using only the line-of-sight velocity dispersion. This ratio is the kinematic correction, which we compute using the tensor virial theorem [e.g., 54, 55]. The effect of flattening on the kinematics of the stars is given by R 3 2 2 d x ρ? (σRR + σφφ ) TRR WRR R = = . (19) 3 2 Tzz Wzz d x ρ? σzz where T and W are the kinetic energy and potential energy tensors [53, 55, 56]. The stellar density in dSphs is well approximated by a Plummer or King profile. Such laws do not lead to tractable integrals in the virial theorem (19). Instead, we approximate the stellar density as a power-law stratified on similar concentric spheroids with m2? = R2 + z 2 q?−2 and so q? is the stellar flattening. This means we can take advantage of equations (19-24) in [55], which give the virial ratios for stellar populations whose density is a pure scale-free power-law declining like

12

1/2 1/2 1 1 + 1 + 8q 2 2

(20)

This formula is given in refs [57, 58]. As is well known, the equipotentials are always rounder than the density contours, so that qφ ≈ 1 even if the dark halo is quite flattened. Then for γ? = 3, we have from [55] √  Q? Q − Q? ArcsinhQ   √ ,    2[ Q? ArcsinhQ − Q]

2 hσxx i = 2 i √  hσzz   Q Q − Q? ArcsinQ   ?√ , 2[ Q? ArcsinQ − Q]

Tensor virial

®

0.4

® ­

0.2 0.0

0.2

γ =2

0.4

γ =4

­

qφ =

γ =3 0.6 2 / σ2 ) log 10 ( σxx zz

distance−γ? . Note that as the all the considered models have infinite mass we must work with the ratios of the velocity dispersions. If we assume the equipotentials are spheroidally stratified, the correction is a function of Q? = qφ2 /q?2 , where qφ is the flattening of the dark halo equipotentials, which is related to the flattening q in the dark halo density via

0.6 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

q

Q? > 1 (21) Q? < 1,

FIG. 5: Kinematic ratio for oblate and prolate figures. Each line shows the prediction from a stellar axisymmetric cusp with density slope γ? flattened with axis ratio q embedded in a halo also with flattening q. The black points show the numerical results from the M2M models.

where (√ Q(Q? ) =



Q? − 1 1 − Q?

Q? > 1 Q? < 1

(22)

Our formulae are appropriate if rt → ∞ or the stellar profile is truncated at a smaller radius than the dark-matter profile. This virial ratio is unity when Q? = 1. This follows because if the stellar density is constant on the equipotentials, then the velocity dispersion is isotropic. It is greater than unity when Q? > 1 (that is, when the model is oblate), as the globally averaged velocity dispersion component along the long or x axis must be larger than that along the short or z axis. It is less than unity when Q? < 1, as the roles of the x and z axes are now reversed for the prolate figure. Formulae for other values of γ? , or stellar density fall-off, are given in Appendix A. We plot the logarithm of the virial ratio in Fig. 5 along with the ratio calculated from the M2M models. The line for γ? = 3 agrees well with the M2M data. The green dashed line shows the virial ratio computed using the virial method. For a stellar density stratified on the same concentric ellipsoids as the dark-matter density, the virial ratio is simply a function of the shape of the ellipsoids and is independent of the radial density profile so the plotted line has a very simple functional form [53, 59, 60]. It can be shown that along the sequence of models the total luminosity-averaged square velocity dispersion σtot is constant. Therefore, for a given model the kinematic factor is the ratio of the total velocity dispersion to the line-of-sight velocity dispersion. When viewed down the x-axis, the kinematic correction factor is Jkin,edge =

 hσ 2 i 2  2 2 hσzz i 2 tot = + 2 2 hσlos i 3 3hσxx i

(23)

This is the ratio of the squared velocity dispersion along the line of sight to the average value. This is smaller (larger) than unity for oblate (prolate) models. When viewed down the z-axis, the kinematic correction factor is Jkin,face =

1 3

+

2 2 2hσxx i 2 i 3hσzz

(24)

This is smaller (larger) than unity for prolate (oblate) models. Note that, as the D-factors are proportional to √ 2 , the D-factor kinematic factor Dkin = Jkin . σtot C.

Flat Rotation Curve Models

A simple but widely-used model of a dark halo has potential-density pair [53, 57] ρDM (R, z) =

2 −2 2 2 2 v02 (2qφ + 1)Rd + R + z (2 − qφ ) , 4πGqφ2 (Rd 2 + R2 + z 2 qφ2 )2

ΦDM (R, z) =

v02 ln(Rd 2 + R2 + z 2 qφ−2 ) 2

(25)

Here, v0 is a velocity scale that is the asymptotic value of the flat rotation curve, whilst Rd is the dark matter length-scale while qφ is the axis ratio of the equipotentials. The dark √ matter density is everywhere positive provided qφ > 1/ 2, so the model can be oblate, spherical or prolate. Unless qφ = 1, the flattening of the dark matter density changes with radius such that the oblate models become more oblate in the outskirts whilst the prolate models become more prolate. At large radii qφ is related to the isodensity flattening q via equation (20).

13 The dark halo is cusped if Rd = 0, but the cusp is isothermal and so much more severe than in the NFW model. The J-factor for the model viewed along the z-axis or symmetry axis is J=

h v04 3(1 − y) − 4qφ2 (y 3 − 1) 3 96Rd D2 G2 qφ

∆21 = qφ2 − q?2 ,

i + qφ4 (8 − 3y − 2y 3 − 3y 5 ) , (26) p with y = Rd / Rd2 + D2 θ2 . At large angles, y → 0 and so the asymptotic value is J→

v04 [3 + 4qφ2 + 8qφ4 ]. 96Rd D2 G2 qφ3

v02 Rd qφ (Dθ/Rd )2 p . GD2 1 + (Dθ/Rd )2

(28)

Note that the total mass of the model is not finite so the D-factor does not tend to a finite value as θ → ∞. Viewed edge-on, two of the integrations for the J-factor are analytic, leaving a final integral over the spherical aperture to be performed numerically J(θ → ∞) = Z ×

v04 1536πRd D2 G2 qφ8



dφ 0

G1 (qφ ) + G2 (qφ ) cos(2φ) + G3 (qφ ) cos(4φ) , (cos2 φ + qφ−2 sin2 φ)3 (29)

G1 (qφ ) = 120 + 280Q2 + 221Q4 + 64Q6 + 9Q8 , G2 (qφ ) = 4Q2 (14 + 3Q2 (3 + Q2 )2 ), 4

2

(30)

4

G3 (qφ ) = Q (7 + 8Q + 3Q ), and Q2 = qφ2 − 1. The D-factor can also be expressed as a single quadrature but the expression is too bulky to present here. Into this dark halo, we embed a population of stars to model the dSph, namely ρ(R, z) =

ρ0 Rc β? . (Rc 2 + R2 + z 2 q?−2 )β? /2

Here, ρ0 is a normalization constant, while q is the axis ratio of the spheroidal isodensity contours. If β? = 5, this is the familiar Plummer model. If the scale-length of the stars Rc is equal to the scale-length of the dark matter Rd , and the flattening of the stellar density q? is equal to the flattening of the dark matter equipotentials qφ , then the phase space distribution function is an isothermal [57]. We derive more general formulae below, but note that this simple limit enables an easy check of the correctness of our results.

∆22 = Rd 2 − Rc 2 ,

D2 = qφ2 Rd 2 − q?2 Rc 2 . (31)

Then the velocity dispersions are 2 hσRR i

v 2 qφ Rc 2 h = 04 4 2 2∆41 Rd 3 DArccosh ∆1 ∆2 D − ∆1 D

(27)

Similarly, the D-factor for a model viewed along the zaxis is given by D=

As both the density and the potential are simple, we can calculate the velocity dispersions seen on viewing the stellar distribution along the short or long axis. We give the results for β? = 5 here, and delegate other formulae to the Appendix B. We begin by defining

2

(2qφ2 Rd 2

+



q φ Rd q? Rc



q?2 (Rc 2 −3Rd 2 ))Arccosh

qφ q?



qφ q?





i − qφ ∆21 ∆22 D2 , v02 q?2 Rc 2 h 2 2 2 2 q ∆ ∆ D − qφ ∆1 D4 Arccosh ∆41 ∆22 D2 ? 1 2   qφ Rd i . + qφ ∆41 DRd 3 Arccosh q? Rc

2 hσzz i =



(32)

The formulae hold generally on using the identity (for S < 1) ArccoshS ≡ −i arccos S These formulae give the line of sight velocity dispersion of an axisymmetric Plummer model viewed along the short and long axes in a dark halo of arbitrary flattening and length-scale. If Rc = Rd , then h v02 qφ 2 5q?4 qφ − 7q?2 qφ3 + 2qφ5 hσRR i = 2 4(qφ − q?2 )3  i qφ 4 2 2 1/2 , + 3q? (qφ − q? ) Arccosh q h v02 q?2 2 hσzz i = q 4 + qφ2 q?2 − 2q?4 2(qφ2 − q?2 )3 φ  i qφ . (33) − 3q?2 qφ (qφ2 − q?2 )1/2 Arccosh q If additionally q? = qφ , then 2 hσRR i=

2v02 , 5

2 hσzz i=

v02 . 5

(34)

With the line of sight velocity dispersion in hand, we can simply rescale the model so that the J-factors are computed for models with the same observables (line of sight velocity dispersion and half-light radius) as the flattenings and the ratio of dark to luminous scalelength Rc /Rd varies. D.

Comparisons

The models in Sections IV B and IV C are complementary. The infinite cusps allow us to vary the central slope of the dark matter. The cored models allow

14 !

!

q η fit

q η fit

Cusp, γ = 3 Core, β = 5, Rd /Rc = 20

Virial Method

Cusp, γ = 3 Core, β = 5, Rd /Rc = 20

Virial Method

1.0 Viewed face-on

Viewed face-on 0.4 0.2

0.0

Rd

FD

FJ

0.5

/R

c=

0.5

20

0

1.0 0.0

0.5

1.0

1.5

c =2

γ =2

γ = 4

2.5

Rd /

Rc

0.2

Rd /R

2.0

0.0

3.0

3.5

=2

4.0

0.0

0.5

1.0

1.5

q η fit

γDM = 0.

5

0.5

.5

γ DM = 0

FD

Rd /Rc = 200 FJ

3.0

3.5

4.0

1.0 Viewed edge-on

0.5

γDM = 1. 5

2 Rc =

R d/

0.5

2.5

Cusp, γDM = 1, γ = 3 Core, β = 5, Rd /Rc = 20

Virial Method

1.0 Viewed edge-on

0.0

2.0

q Cusp, γDM = 1, γ = 3 Core, β = 5, Rd /Rc = 20

Virial Method

00

Rd /Rc = 2

0.4

q q η fit

γ =2 γ =4

0.0

γDM = 1. 5 R /R = 200 d c Rd /Rc = 2

0.5

1.0

1.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

q FIG. 6: J-factor correction factors for oblate and prolate figures viewed face-on (top) and edge-on (bottom). The black points show the numerical results from the M2M models of stellar Plummer models flattened with axis ratio q embedded in NFW dark-matter halos also of axis ratio q. The dashed green line shows the results of the virial method of Section IV A. The blue band shows a range of axisymmetric cusp models from Section IV B. The central line corresponds to a model with γDM = 1, γ? = 3. In the top panel, we have varied the slope of the light profile (note the face-on correction factor is independent of γDM in this case). In the bottom panel, we have varied the slope of the dark matter. In both panels, the red band shows a series of cored flat rotation curve models from Section IV C. The central line has outer stellar density profile of β? = 5 and ratio of dark-matter to stellar scale radii of Rd /Rc = 20. The band corresponds to varying this scale radii ratio by a factor of ten.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

q FIG. 7: D-factor correction factors for oblate and prolate figures viewed face-on (top) and edge-on (bottom). See the caption of Figure 6 for details on each line.

us to vary the ratio of the luminous to the dark matter length-scale. Taken together, a gamut of possibilities of dark halo cusps, density profiles and length-scales can be swept out. The base-10 logarithms of the correction factors (6) are plotted as a function of dark halo flattening q in Fig. 6. The flattening of the stellar population q? is the same as that of the dark halo q. For all plots we use the observed parameters of Reticulum II with a beam angle of 0.5◦ . The correction factors are a function of the beam angle but we have found that this dependence is very weak and that the correction factors are essentially independent of the size of the dSph with respect to the beam size. Therefore, the reported correction factors are appropriate for all beam angles. We show the correction factors for models viewed face-

15 on (top panels) and edge-on (bottom panels). Any correction factor between these extremes should be possible as the inclination angle can be varied between these extremes. The correction factors computed from the M2M Plummer models in NFW halos with axis ratio q are shown as black points. They are in good agreement with the results of the virial method of Section IV A applied to the self-same model, which are shown as a green curve. Note that the exception to this is the q = 2.5 prolate model. As noted in Section II, the Made-to-Measure method in this case produces an equilibrium figure that significantly deviates from a spheroidally stratified Plummer model. We also show as blue bands the range of results for the axisymmetric cusps of Section IV B, in which both the slopes of the dark matter and stellar cusps are allowed to vary. Finally, the red band shows a series of cored flat rotation curve models from Section IV C. The central line has outer stellar density slope of β? = 5 and ratio of dark-matter to stellar scale radii of Rd /Rc = 20. The band corresponds to varying this scale radii ratio by a factor of ten. As we move along the curves, the models have the same line-of-sight velocity dispersion and the same half-light radius. As the models make different assumptions as to the dark matter density and potential, we do not expect these curves to match up exactly with the M2M models, but it is encouraging that they all show similar trends. When an oblate model is viewed along the short axis or face-on, it appears circular, but there is always a boost to the J-factor. For flattenings of q = q? = 0.5, this can be a factor of 3 boost over the spherical J-factor. When the model is viewed edge-on, or along the long axis, then it appears flattened with isophotes of ellipticity 1 − q? . However, the geometric and kinematic corrections work in different directions, the former to boost, the latter to reduce, the J-factor. The net effect is less significant than in the face-on case and is ∼ 0.1−0.2 dex for 0.4 < q < 0.7. When a prolate model is viewed along the long or zaxis, it appears round. Here, the geometric and the kinematic corrections both diminish the J-factor. Although the model looks round on the sky, its J-factor can be substantially less than computed by a spherical analysis. For example, if the true flattening is q = q? = 2.5, then the J-factor is decreased by a factor . 0.3. When the model is viewed edge-on, the isophotes have ellipticity 1 − q?−1 , and the kinematic and geometric factors act in opposite directions with the net result being a small boost. If the flattening is q = q? = 2.5 the J-factor is increased by ∼ 0.3 dex over the spherical estimate. The blue bands give an indication of how the correction factors vary as the dark-matter density slope is adjusted. We find steeper cusps give smaller corrections for models viewed edge-on, but has no effect on the corrections for models viewed face-on as the geometric factor is independent of γDM . For the face-on case we see that making the slope of the stellar density profile steeper produces larger corrections to the J-factor. From the red bands we observe that making the dark

matter halo more extended (increasing Rd /Rc ) produces larger corrections for the face-on case but smaller corrections for the edge-on case. The width of the red bands when the length-scales Rc /Rd is varied is at most 0.5 dex, even at the most extreme flattenings. Most of the Milky Way dSphs are rounder than q? = 0.5. In this regime, the red band is thinner, and gives rise to an uncertainty of at most ∼ 0.25 dex. This suggests that varying the concentration of the dark matter halo will not have a significant effect on the flattened J-factors. This is corroborated by experiments with the Plummer profile embedded in the NFW profile. Additionally, making the stellar density in these models fall off more rapidly increases the magnitude of the J-factor correction factors when viewing face-on but decreases the magnitude of the correction when viewing edge-on. The equivalent results for the D-factors are shown in Fig. 7. The correction factors for the M2M models models are all . 0.2 dex, and suggest that for most applications, the spherical approximation suffices for the Dfactors. Note that in the face-on case, the cored models give a similar approximation of the correction factors as the cuspy models whilst for the edge-on case the cored models more faithfully represent the true correction factors than the cuspy models. Increasing the outer stellar density slope for the flat rotation curve models increases the magnitude of D-factor correction factors when viewing face-on but has little effect for the edge-on case. Finally, we note that for an (α, β, γ) stellar model of equation (3) embedded in another (α, β, γ) dark-matter model the J-factor correction factors are very insensitive to the choice of the density slopes of the stellar and dark-matter distributions and the ratio of the stellar to dark-matter scale-lengths. The same is broadly true for the D-factor correction factors except that the edge-on correction factor has a weak dependence with the outer slope of the dark matter profile. This is slightly at odds with the flat rotation curve model results but this may be due to the flat rotation curve models having a density flattening that varies with radius whilst the (α, β, γ) models have a constant density flattening.

V.

THE EFFECTS OF TRIAXIALITY

Generically, we might expect the light and dark-matter distributions in dwarf spheroidals to be triaxial [61]. Triaxiality can introduce additional flattening (stretching) along the line-of-sight and so naturally increases (decreases) the J-factor and gives rise to larger (smaller) correction factors. Here we extend the formulae given in the previous section to account for intrinsic triaxial shapes. We begin by focusing on the infinite cusp models where some analytic progress can be made before moving on to consider more general density profiles. We extend the models of equation (11) and introduce an intermediate-to-major axis ratio p in addition to the minor-to-major axis ratio q. Here we restrict q < p < 1

16 such that a prolate model has p = q = 6 1. It is conventional to use a triaxiality parameter T to describe the figures defined by T =

1 − p2 . 1 − q2

(35)

Note that figures with T = 0 are oblate whilst those with T = 1 are prolate. The density for the triaxial cusp models is

ρDM (m) =

3 − γDM Mh 3−γDM mγDM 4πpqmh

for m ≤ rt , (36)

and zero otherwise. m2 = x2 + y 2 p−2 + z 2 q −2 and rt is a truncation ellipsoidal radius. When an infinite (rt → ∞) model is viewed along the z axis, the observed flattening is p and the geometric factor is a combination of equation (15) and (16) such that Z p2−γDM 2π Jgeo,z = dθ(cos2 θ + p−2 sin2 θ)1/2−γDM . 2πp2 q 0 (37) In this case, the observed major-axis length corresponds to the intrinsic model scale radius. For γDM ≤ 2, the integral is a monotonic function of q that is greater than unity for q < 1 and less than unity for q > 1. If viewed along the y-axis the observed flattening is q and the Jfactor is given by

If the infinite model is observed along a line of sight oriented with spherical polar angles (ϕ, ϑ) with respect to the intrinsic Cartesian coordinates of the model, the geometric factor must be computed with the full threedimensional integrals as Z Dα Z 2π Z ∞ 1 0 dR RρDM 2 (x), dθ dz Jgeo = Jsph D2 −∞ 0 0 (41) ˆ the ˆ + R sin θϑˆ + z 0 rˆ with (ˆ ˆ ϑ) where x = R cos θϕ r , ϕ, set of spherical polar unit vectors. Making the model finite with rt < Dθ produces J/Jsph = 1/(pq). However, for this general viewing angle calculation of the observed scale radius seems intractable. The kinematic fac2 2 tors Jkin = (hσtot i/hσlos i)2 can be derived for these more general viewing angles as h1 i2 1 + f1 + f2 Jkin = , 2 2 2 3 cos2 ϑ + f1 sin ϑ cos2 ϕ + f2 sin ϑ sin ϕ (42) where Rπ R 2π 2 dθ 0 dφ F (θ, φ) sin3 θ cos2 φ i hσxx 0 f1 = 2 = R π > f2 R 2π hσzz i dθ 0 dφ F (θ, φ) sin θ cos2 θ 0 Rπ R 2π 2 hσyy i dθ 0 dφ F (θ, φ) sin3 θ sin2 φ 0 f2 = 2 = R π > 1, R 2π hσzz i dθ 0 dφ F (θ, φ) sin θ cos2 θ 0 (43) and

2−γDM



F (θ, φ) = (sin2 θ cos2 φ+P?2 sin2 θ sin2 φ+Q2? cos2 θ)−γDM /2 . (44) 0 (38) Here P? = pφ /p and Q? = qφ /q with pφ and qφ being and again the observed major-axis length coincides with the axis ratios of the dark matter potential. We see that the intrinsic model scale radius. When viewed along the when viewing down the major axis (ϑ = π/2, ϕ = 0) major-axis, the observed flattening is q/p and the obthe kinematic correction factor is less than unity whilst served major-axis length coincides with the intermediviewing down the minor axis (ϑ = 0, ϕ = 0) produces ate axis so the resultant measured scale-length must be a kinematic correction factor greater than unity. Genscaled by a factor 1/p. This gives rise to a geometric erally, we find that Jkin,x < Jkin,y < Jkin,z such that factor of the total correction factors for γDM < 3/2 obey the hiZ erarchy FJ,x < FJ,y < FJ,z . We have found that the (q/p)2−γDM 2π 2 2 −2 1/2−γDM effects of triaxiality seem to be in accordance with our Jgeo,x = dθ(cos θ+(q/p) sin θ) . 2π(q/p)2 p 0 expectation from the axisymmetric case. When there is (39) additional flattening along the line-of-sight the geometric As with the axisymmetric case, the infinite cusps have and kinematic correction factors combine to increase the limited use and it is more practical to use models with correction factor, whilst additional stretching decreases finite truncation ellipsoidal radii rt < Dθ. In this case, the correction factor. the geometric factors are given by We now compute general triaxial correction factors using the method of Section IV A. We show an example of 1−γDM Jgeo,x = (qp) , the J-factor correction factors for the Reticulum II model 1−γDM (40) Jgeo,y = q /p, presented in Section II but with stellar minor-to-major axis ratio q = 0.4 and stellar intermediate-to-major axis Jgeo,z = p1−γDM /q. ratio p = 0.73. We assume the dark-matter distribution For the models that produce a finite J-factor (γDM < is flattened in the same way as the stellar distribution. 3/2), we find Jgeo,x < Jgeo,y < Jgeo,z . For the This model has triaxiality parameter T = 0.55, (which astrophysically-motivated case of γDM = 1 the geometwas deemed the best-fit to the Local Group dSphs by ric factors are simply Jgeo,x = 1, Jgeo,y = 1/p and [61]). The base-10 logarithm of the correction factor for Jgeo,z = 1/q. all viewing angles is given in Fig. 8. We see that, in Jgeo,y =

q 2πq 2 p

Z

dθ(cos2 θ + q −2 sin2 θ)1/2−γDM ,

17 VI. INTRINSIC SHAPES AND AXIS ALIGNMENTS OF DWARF SPHEROIDAL GALAXIES

FIG. 8: J-factor correction factors for a triaxial model of Reticulum II: each point on the sphere is colored by the correction factor when viewing the model along the radial vector that passes through that point. The black contours show the observed ellipticity when viewed from that direction. The small ellipsoid shows an isodensity contour for the considered model which has axis ratios p = 0.73 and q = 0.4 in both the stellar and dark-matter distributions. The largest correction factor is achieved when viewing the model down the short axis (z) whilst the smallest correction factor is achieved when viewing the model down the long axis (x). When viewing down the intermediate axis (y) the observed ellipticity matches that of Reticulum II.

agreement with the simple predictions from the infinite cusp models, the largest correction factor occurs when the model is viewed down the short axis (the z axis) whilst the smallest is when viewing down the long axis (the x axis). The black contours on the sphere show lines of constant observed ellipticity. We see that for this figure an observed ellipticity of 0.3 gives rise to a variation in the correction factor of 0.6 dex. In conclusion, additional flattening along the line-ofsight can lead to an increase in the correction factors. For a general triaxial figure the largest correction factor is obtained when viewing the model down the short axis whilst the smallest correction factor is yielded when viewing the model down the long axis.

We have presented corrections to the J- and D-factors based on the assumption that the dSphs are prolate or oblate figures with axes aligned with the line-of-sight. Such configurations are quite unlikely as we anticipate that generically the dSph principal axes are misaligned with the line-of-sight. In this section we will discuss what is known regarding the intrinsic shapes of the dSphs and how this translates into observed properties via their axes alignment with respect to the line-of-sight. For a given individual galaxy, we have a couple of probes of its intrinsic shape [62–64]. The first of these is the presence of isophotal twisting, that is the change in the orientation of the major axis of the isodensity contours with on-sky distance from the galaxy center. Isophotal twisting is a clear signature of a triaxial figure with varying axis ratios with radius, although isophotal twisting may also be caused by tidal effects [65]. Another indicator of triaxiality is evidence of kinematic misalignment between the axis of rotation and the minor axis of the projected density. For entire populations of galaxies, progress can be made by analyzing statistics of the population [e.g. 50]. Recently, [61] demonstrated that under the assumption that the intrinsic axes of the dSphs are randomly oriented, the Local Group dSph population is best reproduced by triaxial models with mean triaxiality T¯ = 0.55+0.21 −0.22 and a mean intrinsic ellipticity (E = 1 − (c/a)) ¯ = 0.51+0.07 . The assumption of random orientation of E −0.06 is perhaps to be questioned, particularly for the Milky Way dSphs. Dark-matter only simulations [66, 67] have demonstrated that the major axes of subhalos tend to be aligned with the radial direction to the center of their host halo and this picture has been corroborated when baryons have been included [68]. The main exception to this is near the subhalo’s pericentric passage where the major axis is briefly aligned perpendicular to the radial direction. Most of the dSphs are distant enough for the radial direction and our line-of-sight to approximately coincide, which suggests that for many dSphs the observed flattening corresponds to the intermediate-to-minor axis ratio and that there is significant stretching of the dSphs along the line-of-sight. As demonstrated in this paper, this gives rise to overestimates of the J-factors from spherical analyses for the prolate face-on models and for the triaxial model viewed down the major axis. Based on this discussion, we now compute the expected J correction factors with their associated uncertainties under a number of assumptions regarding the intrinsic shape and alignment of the dSphs. We use the emcee package from [69] to draw 500 samples of (T, E, ϑ, ϕ) i.e. the triaxiality, the intrinsic ellipticity and the two viewing angles. Our likelihood is the distribution of the observed ellipticity for each dSph given by, for instance, equation

18

1. Uniform (U): T ∼ U(0, 1), E ∼ U(0, 0.95), cos ϑ ∼ U(0, 1), ϕ ∼ U(0, π/2), 2. Viewing down the major-axis (R): T ∼ U(0, 1), E ∼ U(0, 0.95), ϑ ∼ N (π/2, 0.1 rad), ϕ ∼ N (0, 0.1 rad), 3. S´ anchez-Janssen et al. [61] priors (T): T ∼ N (0.55, 0.04), E ∼ N (0.51, 0.12), cos ϑ ∼ U(0, 1), ϕ ∼ U(0, π/2), where U(a, b) is a uniform distribution from a to b and N (µ, σ) is a normal distribution with mean µ and standard deviation σ. For each sample we compute the base10 logarithm of the correction factor FJ to construct a distribution of correction factors. In Figure 9, we show the full 1D distributions of the correction factors for Reticulum II. All three prior assumptions produce a correction factor distribution that peaks near zero. The broadest distribution corresponds to the case where uniform priors have been adopted in all parameters. In this case, the largest correction factors correspond to models with high intrinsic ellipticity E viewed down the short axis. These models have triaxiality T close to unity so are near prolate models. The smallest correction factors correspond to models with low intrinsic ellipticity viewed down the long-axis. For the prior assumption that we are viewing along the major axis, we find the median correction factor peaks at ∼ −0.2dex. For this prior assumption, there is an approximate one-to-one relationship between T and E as well as T and FJ . Models with smaller T correspond to smaller E and hence smaller amplitude correction factors as these models are approximately edge-on oblate, whilst larger T and larger E produce larger amplitude negative correction factors as these models are nearer prolate stretched along the line-of-sight. For the prior assumption that the models have some fixed triaxiality and intrinsic ellipticity, the largest correction factors correspond to viewing angles nearer the short axis and the smallest correction factors correspond to viewing angles closer to the long axis. The medians and ±1σ error-bars of the correction factors for all the dSphs computed for the three prior assumptions are given in Table V. This information is also displayed in Figure 10. If we assume the dSphs are preferentially viewed along the major axis, the median correction factor is less than unity and is weakly decreasing with increasing ellipticity. As the dSphs become more flattened on the sky, they are forced to become more extended along the line-of-sight and so the J-factor decreases. The medians of the correction factors for dSphs with small ellipticity under the uniform prior assumption is around 0.05 dex and the upper error-bars are in general

6

Uniform (U) Major axis (R) SJ 2016 (T)

5 4

dN/dF J

(A1,A2) of [50]. For those dSphs with upper-bounds on their ellipticity we use a normal distribution with mean zero and standard deviation of half the upper-bound. We consider three different prior distributions on the parameters (T, E, ϑ, ϕ):

3 2 1 0 0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

FJ

FIG. 9: Distribution of the base-10 logarithm of the J-factor correction factors for triaxial models of Reticulum II under three different assumptions on the prior distributions of the intrinsic triaxiality, ellipticity and viewing angles as described in the text.

larger than the lower error-bars but the spread encompasses zero. The median shift is due to the asymmetry in the correction factors between oblate and prolate models seen in Fig. 6. This suggests that all J-factors are underestimated by ∼ 10 percent but naturally this conclusion is very sensitive to the exact prior assumptions. For small ellipticity, the spread in the correction factors for the uniform and fixed shape priors are approximately equal with the spread weakly increasing with increasing ellipticity for the uniform case. For the fixed shape prior the spread decreases at large ellipticity as models viewed down the minor axis become inconsistent with the observed ellipticity. We have fitted a simple relation to the uncertainty in the correction factors from the uniform priors ∆FJ (the average of the ±1σ uncertainties) as a function of ellipticity as √ ∆FJ ≈ 0.4 , (45) which gives a fractional uncertainty in the J-factor of √ ∆J ≈ 100.4  − 1. J

(46)

This expression gives 50 per cent uncertainty for  ≈ 0.2, a factor 1.8 uncertainty for  ≈ 0.4 and a factor 2.3 uncertainty for  ≈ 0.6. For small , ∆J/J ≈ 0.9. We conclude for a typical dSph ellipticity of 0.4 there is approximately a factor of two uncertainty in the J-factors due to the unknown triaxiality and alignment of the dSph.

VII.

DISCUSSION AND CONCLUSIONS

Flattening is a crucial attribute of a dwarf spheroidal galaxy. Both the dark halo and the stellar distribution

19 TABLE V: Median and ±1σ J correction factors for the known dSphs for three different assumptions about the intrinsic shapes and alignments. The first correction factor (marked ‘U’) uses flat uniform priors on the triaxiality, minor-to-major axis ratio and viewing angles. The second (marked ‘R’) uses a normal prior on the viewing angles centered on (θ = π/2, φ = 0) with width 5◦ . The third (marked ‘T’) uses a normal prior on the triaxiality and minor-to-major axis ratios with means (0.55, 0.49) and widths (0.04, 0.12) (based on the fits to the Local Group dSphs of S´ anchez-Janssen et al. [61]. Name Hydra II Leo T Leo II Segue 2 Leo I Horologium I Fornax Draco Sculptor Carina Sextans Coma Berenices Bo¨ otes I Canes Venatici I Tucana II Pisces II Grus I Willman 1 Segue 1 Leo IV Leo V Canes Venatici II Ursa Minor Reticulum II Ursa Major II Hercules Ursa Major I

Ellipticity 0.01+0.20 −0.01 < 0.10 0.13+0.05 −0.05 0.15+0.10 −0.10 0.21+0.03 −0.03 < 0.28 0.30+0.01 −0.01 0.31+0.02 −0.02 0.32+0.03 −0.03 0.33+0.05 −0.05 0.35+0.05 −0.05 0.38+0.14 −0.14 0.39+0.06 −0.06 0.39+0.03 −0.03 0.39+0.10 −0.20 0.40+0.10 −0.10 0.41+0.20 −0.28 0.47+0.08 −0.08 0.48+0.13 −0.13 0.49+0.11 −0.11 0.50+0.15 −0.15 0.52+0.11 −0.11 0.56+0.05 −0.05 +0.02 0.59−0.03 0.63+0.05 −0.05 0.68+0.08 −0.08 0.80+0.04 −0.04

FJ,U −0.01+0.10 −0.02 −0.00+0.07 −0.02 0.02+0.18 −0.07 0.01+0.25 −0.07 0.03+0.28 −0.11 0.00+0.13 −0.04 0.04+0.38 −0.13 0.04+0.28 −0.13 0.04+0.36 −0.15 0.06+0.31 −0.15 0.06+0.27 −0.16 0.04+0.32 −0.15 0.04+0.42 −0.15 0.07+0.27 −0.17 0.03+0.33 −0.13 0.07+0.36 −0.18 0.01+0.29 −0.12 0.08+0.33 −0.21 0.05+0.35 −0.18 0.07+0.32 −0.20 0.05+0.33 −0.18 0.09+0.33 −0.22 0.03+0.41 −0.18 +0.29 0.08−0.23 0.06+0.32 −0.22 0.08+0.35 −0.23 0.02+0.39 −0.19

can be flattened. The ultrafaint dSphs have many fewer baryons than the classical dwarfs so it is anticipated that feedback effects have a weaker effect on the shape of the dark matter distribution in the ultrafaints. Therefore, for the ultrafaints, a flattened stellar distribution probably corresponds to a flattened dark matter distribution. Of these ultrafaints, Reticulum II is an interesting object as it is particularly nearby and also one of the most highly flattened of all the ultrafaints, at least as judged by the stellar light. On these grounds, we might well expect that flattening may provide an explanation as to why a gamma-ray signal may have been seen towards Reticulum II as opposed to other ultrafaints. We have explored the impact of flattening on the J- and D-factors, which control the expected dark matter annihilation and decay signals from the dSphs. The effects of flattening on these factors can be decomposed into two separate corrections: the geometric and the kinematic factors. The first of these corresponds to the increase (decrease) in dark-matter density produced by squeezing (stretching) the models. The latter corresponds to how the observed velocity dispersion relates to the total velocity dispersion or the enclosed dark matter mass. When viewing oblate (prolate) models face-on, these two fac-

FJ,R −0.05+0.04 −0.12 −0.06+0.04 −0.13 −0.11+0.06 −0.14 −0.11+0.07 −0.13 −0.16+0.07 −0.15 −0.08+0.06 −0.13 −0.19+0.07 −0.15 −0.19+0.06 −0.13 −0.21+0.08 −0.13 −0.20+0.08 −0.13 −0.21+0.08 −0.12 −0.21+0.08 −0.13 −0.22+0.07 −0.12 −0.22+0.07 −0.13 −0.20+0.09 −0.14 −0.20+0.07 −0.12 −0.21+0.11 −0.12 −0.24+0.07 −0.12 −0.23+0.06 −0.12 −0.24+0.07 −0.11 −0.24+0.08 −0.11 −0.23+0.06 −0.11 −0.25+0.06 −0.12 +0.07 −0.26−0.10 −0.26+0.06 −0.09 −0.24+0.06 −0.09 −0.22+0.08 −0.08

FJ,T 0.07+0.19 −0.17 0.03+0.08 −0.06 0.09+0.15 −0.15 0.09+0.21 −0.17 0.13+0.20 −0.23 0.09+0.19 −0.15 0.06+0.33 −0.21 0.05+0.30 −0.17 0.04+0.28 −0.19 0.03+0.27 −0.18 0.00+0.24 −0.16 0.03+0.28 −0.16 0.00+0.24 −0.18 −0.01+0.17 −0.14 0.01+0.22 −0.14 0.00+0.21 −0.16 0.01+0.24 −0.14 −0.01+0.16 −0.13 −0.01+0.19 −0.16 −0.02+0.15 −0.15 −0.01+0.20 −0.15 −0.02+0.15 −0.13 −0.03+0.08 −0.11 +0.08 −0.04−0.12 −0.04+0.07 −0.13 −0.05+0.07 −0.10 −0.06+0.06 −0.10

tors act together to increase (decrease) the J-factor over a spherical analysis, whereas, when viewing these models edge-on, the two factors compete and result in a decrease (increase) in the J-factor over a spherical analysis. We have used Made-to-Measure techniques [38, 39] to build numerical equilibrium models of Reticulum II. These reproduce the flattened shape, the major-axis length and the line of sight velocity dispersion of Reticulum II. For the models with a prolate dark matter halo with ellipticity ∼ 0.6 viewed edge-on, flattening could cause an additional amplification of ∼ 2 − 2.5 for Reticulum II over that expected for spherical dark halos. This factor could be still larger if the stellar profile falls off more slowly than a Plummer law (which could increase the kinematic factor). It could also be larger if the dark halo of Reticulum II is triaxial (as anticipated from darkmatter-only simulations) and hence more flattened along the line-of-sight. However, this scenario is disfavored by dark-matter simulations with and without baryons that produce subhalos which preferentially point towards the center of their host halo and so we might anticipate dSphs to be elongated along the line-of-sight. We corroborated the results of the Made-to-Measure simulations with a simpler virial method that allows for

20

0.8

Uniform (U)

Major axis (R)

SJ 2016 (T)

0.6

FJ

0.4 0.2 0.0 0.2

Hydra II Leo T Leo II Segue 2 Leo I Horologium I Fornax Draco Sculptor Carina Sextans Coma Berenices Tucana II Canes Venatici I Boötes I Pisces II Grus I Willman 1 Segue 1 Leo IV Leo V Canes Venatici II Ursa Minor Reticulum II Ursa Major II Hercules Ursa Major I

0.4

FIG. 10: Medians and ±1σ error-bars for the base-10 logarithms of the J-factor correction factors for all the dSphs ranked by their ellipticity. The three different error-bars correspond to three different prior assumptions regarding the intrinsic shapes and alignments of the dSphs as described in the text.

more rapid calculation of the correction factors for general geometries. A simple fitting relation has been provided for rapid estimation of the correction factors for the oblate and prolate cases. Additionally, we have inspected two cases where some analytic progress can be made in the computation of the J-factors. This has allowed us to characterize how the correction factors change as a function of the stellar and dark-matter distributions. We found that the correction factors for the Made-toMeasure models agree well with the trends seen in the analytic models. We used our models to estimate the J-factors for the dSphs under the assumption that the figures are aligned with the line-of-sight and are either oblate or prolate. The ranking of the J-factors of the dSphs is slightly altered when accounting for flattening under the assumption that all the dSphs are either prolate or oblate. Typical correction factors for a dSph with ellipticity 0.4 are 0.75 in the oblate case and 1.6 in the prolate case. We also demonstrated that the corrections to the D-factors are much smaller than the scatter in the spherical Dfactor from the other observables. For instance, a dSph with ellipticity 0.4 has a D-factor correction factor of 0.97 in the oblate case and 1.1 in the prolate case. Therefore, we concluded that flattening is unimportant for D-factor computation. We concluded our discussion of the effects of flatten-

ing by computing correction factors for triaxial figures. The findings from the axisymmetric cases were found to simply extend when considering triaxiality. The largest J-factor correction factor corresponds to viewing the figure along the minor axis, whilst the smallest corresponds to viewing the figure along the major axis. We found that for a Reticulum II-like model the J-factor correction factor varies by a factor of ∼ 6 − 10 as one changes the viewing angle. For a fixed observed ellipticity, the correction factor can vary by a factor of ∼ 4. We demonstrated that for the known dSphs the uncertainty in the correction factors due to unknown triaxiality increases with the observed ellipticity of the dSph and is typically a factor of two for  ∼ 0.4. If all dSphs have their major axes aligned with the line-of-sight (as suggested by some simulations), the correction factors decrease as a function of observed ellipticity and are typically a factor 1/2 for 0.4 .  . 0.6. Deviations from sphericity in both the light profile and the dark matter are important. This suggests fundamental limitations to the spherical Jeans modeling which is common in the field (although see [32] for Jfactors computed using axisymmetric Jeans modeling). In particular, increasingly sophisticated statistical techniques [70, 71] will fail to include an inherent uncertainty if the assumption of a spherical stellar density profiles in a spherical dark halo breaks down. The uncertainties,

21 which are different for different dSphs, must be accounted for joint analyses of multiple dSphs. Spherical Jeans modeling is probably most useful for large classical dwarf spheroidal galaxies that look nearly round (such as Leo I or Fornax). It ignores important uncertainties for the ultrafaints, which is unfortunate as these are the most promising targets of all for indirect dark matter detection. We hope that the work presented here – a systematic foray into the domain of flattening – is the beginning of a systematic exploration of more general flattened and triaxial dark halo shapes. Finally, this study was partly inspired by the gammaray detection [17] toward the very flattened ultrafaint, Reticulum II. Our work demonstrates that Reticulum II could have a J-factor that is higher than spherical analyses suggest if it is a prolate figure. However, the correction for the prolate shape does not make Reticulum II stand out as the dSph with the highest J-factor nor does a lack of signal from the other dSphs create any tension, irrespective of the shapes of the other dSphs. If, however, Reticulum II is an oblate figure the J-factor is lower than that found through spherical analyses and lack of signal from the other dSphs may give rise to some tension if

the other dSphs (such as Ursa Minor) are prolate. More generically we have demonstrated that unknown triaxiality produces uncertainty in the J-factor for Reticulum II of a factor of ∼ 2. In general, we have shown that the effect of flattening on expected dark matter annihilation fluxes cannot be ignored. Indeed, flattening can shift expected signals by amounts larger than error bars due to current velocity dispersion measurements. These currently unknown shifts change the ranking of dSph targets for gamma-ray experiments. However, if the orientations of the Milky Way dSphs can be determined, the results presented here can help pin down relative J-factors and allow tests of dark matter explanations of gamma-ray detections.

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Acknowledgments

JLS acknowledges financial support from the Science and Technology Facilities Council (STFC) of the United Kingdom. Figure 1 was produced using the Pynbody package [72].

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Appendix A: Virial ratios for cusped models

In Section IV B we presented the virial ratio 2 2 hσxx i/hσzz i for a flattened cusped model with density slope γ? = 3. Here we provide formulae for other values

23 so that when q? = qφ , we obtain

of γ? . For γ? = 2, the virial ratio is 2 Q? T (Q) − Q hσxx i = , 2 i hσzz 2[Q − T (Q)]

(A1)

where T = Arctanh for Q? < 1 and T = Arctan for Q? > 1, and Q is defined in equation (22). For γ? = 4, the virial ratio is 2 hσxx i Q? [(Q2 − 1)T (Q) + Q] = 2 hσzz i 2[Q? T (Q) − Q]

(A2)

where we have corrected a typographical error in eq. (21) of Agnello and Evans [55]. For γ? = 5, the virial ratio is simply 2 hσxx i = Q? . 2 hσzz i

(A3)

A plot of the ratios as a function of flattening are given as Figure 1 in [55]

2 hσRR i=

v02 , 2

In Section IV C, we presented formulae for the velocity dispersions of a cored stellar profile with outer density slope β? = 5 embedded in a cored dark matter density profile. Here we provide equivalent formulae for the cases β? = 4 and β? = 6. Let us recall the definitions ∆21 = qφ2 − q?2 ,

∆22 = Rd 2 − Rc 2 ,

(B1)

and let us introduce the function     Rd ∆1 Rc ∆1 i 1 h Arctan − Arctan . F = ∆1 ∆2 q? ∆2 qφ ∆ 2 (B2) Notice that if Rc > Rd or q? > q, this function remains well-defined on using the identity (for S < 0) √ √ 1 1 √ Arctan S ≡ √ Arctanh −S. −S S For the case β? = 4, v02 qφ Rc h 2 2 (qφ Rd + q?2 (Rc 2 −2Rd 2 ))F (B3) ∆21 ∆22 i − qφ Rc + q? Rd , i v 2 q 2 Rc h q? 2 qφ F − . (B4) hσzz i = 0 ?2 ∆1 q? Rc + qφ Rd

2 hσRR i =

If Rc = Rd , then the velocity dispersions are a lot simpler. A careful Taylor expansion gives 2 hσRR i = 2 hσzz i =

2qφ (2q? + qφ )v02 , 3(q? + qφ )2 q?2 v02 . (q? + qφ )2

(B5)

v02 . 4

(B6)

Finally, we give the results for β? = 6, v02 qφ Rc 2 h Rd (Rc 2 + 2Rd 2 )∆21 − q? qφ Rc ∆22 ∆21 ∆42 q? Rc + qφ Rd i − Rc (3qφ2 Rd 2 + q?2 (Rc 2 − 4Rd 2 ))F ,

2 hσRR i =

(B7)

2

2 hσzz i =

q? qφ Rc Rd + qφ2 Rd − q?2 ∆22 i v02 q?2 Rc 2 h −qφ Rc F + . 2 2 ∆ 1 ∆2 (q? Rc + qφ Rd )2

If Rc = Rd , then 2 hσRR i = 2v02 2 i = v02 hσzz

Appendix B: Velocity dispersions for flat rotation curve models

2 hσzz i=

qφ (8q?2 + 9q? qφ + 3qφ2 ) , 15(q + qφ )3

q?2 (3q? + qφ ) . 3(q? + qφ )3

(B8)

and if additionally q? = qφ , we recover 2 hσRR i=

v02 , 3

2 hσzz i=

v02 . 6

(B9)

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